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Basic definitions for continued fractions

A continued fraction is an expression of the form

(1)

where , and are either real or complex values called the terms of continued fraction (1). The number of terms can be either finite () or infinite (). If continued fraction (1) is called finite, otherwise infinite. The ’s are called partial denominators and ’s partial numerators.

Thought the notation used for a continued fraction in (1) is intuitive and lucid, it takes up a lot of space and is not easy to typeset. So several alternative notations were developed. For instance, A.I. Pringsheim introduced the notation

while C.F.Gauß used

where K stands for the initial of German mane Kettenbrüche for continued fraction.

The rational value

(2)

or

or

which we get by truncation of the original continued fraction after the th partial denominator , , is called the th convergent of the given continued fraction.

The first four convergents are

(3)

Although a given convergent may naturally be worked out “from the bottom” as it was done above, it is more practical to generate the sequence of convergents “from the top” . Let , denote the numerator and the denominator of the th convergent. J.Wallis [1] , [2] and L.Euler [3] found that the values of , and can be determined recursively

(4)

(5)

for and initial conditions , , , . It is sometimes useful to start with , , , , or with , , , .

Recurrence relations (4) and (5) are called fundamental recurrence formulas and the fraction

Simple continued fractions

If for every , then the continued fraction (1) is called simple. Simple continued fractions

(19)

are usually written using a more compact abbreviated notation

(20)

The standard textbooks usually works only with simple continued fractions, but simple continued fractions are not necessarily “simple”. For instance, presently no regularity in the sequence of partial denominators of the simple continued fraction of π is known

Functional continued fractions

The terms of a continued fraction (1) can also be functions, say , , and , . For instance, to J.H.Lambert [6] , [7] attributed relation

(24)

appears already in L.Euler [3] . Lambert used this continued fraction expansion to prove that is irrational. He showed by an infinite descent argument that if is rational, then the right hand side of (24) is irrational. Since is rational, the conclusion follows (Some authors claim that Lambert's proofs is not complete. Pringsheim [8] proved that this is not true.) Lambert gave a list of the first 27 convergents of the simple continued fraction expansion of from which the first 25 were correct, but the last two not.

For the arc tangent we have

(25)

Ascending continued fractions.

Continued fractions of type (1) are called descending continued fractions. However there are possible also other types of continued fractions. One of them are so-called the ascending continued fractions. They were already developed by Fibonacci, Lambert or Lagrange.